\begin{frame} \frametitle{Disproving Semantic Entailment} \begin{block}{} How to disprove $ \alpha_1,\ldots,\alpha_n \;\models \; \beta$? \pause That is, $ \alpha_1,\ldots,\alpha_n \;\nmodels \; \beta$? \bigskip \pause Find a valuation (assignment of truth values to variables) that \begin{itemize} \item makes $\alpha_1,\ldots,\alpha_n$ true, and \item $\beta$ false. \end{itemize} \end{block} \pause \begin{exampleblock}{} Do we have \quad $p \vee q \;\models\; p \to q$ \quad ? \pause \begin{center} \begin{tabular}{|c|c|c|c|} \hline \thd $p$ & \thd $q$ & \thd $p \vee q$ & \thd $p \to q$ \\ \hline $\F$ & $\F$ & $\F$ & $\T$\\ \hline $\F$ & $\T$ & $\T$ & $\T$\\ \hline $\T$ & $\F$ & \malert{1}{3}{$\T$} & \malert{1}{3}{$\F$}\\ \hline $\T$ & $\T$ & $\T$ & $\T$\\ \hline \end{tabular} \end{center} \pause\pause When $p$ is $\T$ and $q$ is $\F$, then $p \vee q$ is $\T$ and $p \to q$ is $\F$. \pause\medskip Conclusion: $p \vee q \;\not\models\; p \to q$\;. \end{exampleblock} \end{frame}